MGB2: Bivarite MGB2 copula
In lizhengxiao/rMGLReg: MGL copula regressions

Description Usage Arguments Details Value References Examples

Density, distribution function, and h-functions for the bivariate MGB2 copula proposed in Yang et al.,(2011).

dcMGB2.bivar(u1, u2, pars1, pars2, pars3)

pcMGB2.bivar(u1, u2, pars1, pars2, pars3)

hcMGB2.bivar(u1, u2, pars1, pars2, pars3)

`u1, u2`	numeric vectors of equal length with values in [0,1].
`pars1, pars2, pars3`	copula parameters, denoted by p_1, p_2, q.

The MGB2 copula with parameters (p1, p2, q) has joint density

c(u_1,u_2;p_1,p_2,q)=\frac{Γ(q)Γ(∑_{i=1}^2 p_i + q)}{∏_{i=1}^2Γ(p_i+q)}\frac{∏_{i=1}^2 (1 + x(u_i;p_i,q))^{p_i+q}}{(1 + ∑_{i=1}^{2}x(u_i;p_i,q))^{∑_{i=1}^{2}p_i+q}},

for p_1, p_2>0, q>0. (Here Gamma(a) is the function implemented by R's gamma and defined in its help.

The joint cdf of the MGB2 copula is

C(u_1,u_2;p_1,p_2,q)=\int_{0}^{+∞}∏_{i=1}^{2}G_p(\frac{I_{p_i,q}^{-1}(u_i)}{(1-I_{p_i,q}^{-1}(u_i))θ})\times \frac{θ^{-(q+1)}e^{-1/θ}}{Γ(q)}dθ,

where I_{m,n}^{-1}() denotes the inverse of the beta cumulative distribution function (or regularized incomplete beta function) with parameters shape1 = m and shape2 = n implemented by R's qbeta.

The h-function is defined as the conditional distribution function of a bivariate copula, i.e.,

h_1(u_2|u_1,p_1,p_2,q) := P(U_2 ≤q u_2 | U_1 = u_1) = \partial C(u_1,u_2) / \partial u_1,

h_2(u_1|u_2,p_1,p_2,q) := P(U_1 ≤q u_1 | U_2 = u_2) := \partial C(u_1,u_2) / \partial u_2,

where (U_1, U_2) \sim C, and C is a bivariate copula distribution function with parameter(s) p_1,p_2,q.

dcMGB2.bivar gives the density

pcMGB2.bivar gives the distribution function

hcMGB2.bivar gives the h-functions (hfunc1, hfunc2).

Xipei Yang, Edward W Frees, and Zhengjun Zhang. A generalized beta copula with applications in modeling multivariate long-tailed data. Insurance: Mathematics and Economics, 49(2):265-284, 2011.

dcMGB2.bivar(u1 = c(0.2, 0.5), u2 = c(0.9, 0.2), pars1 = 0.5, pars2 = 0.5, pars3 = 1.5)
pcMGB2.bivar(u1 = c(0.5, 0.1), u2 = c(0.9, 0.1), pars1 = 0.5, pars2 = 0.5, pars3 = 1.5)
hcMGB2.bivar(u1 = c(0.5, 0.1), u2 = c(0.9, 0.1), pars1 = 0.5, pars2 = 0.5, pars3 = 1.5)$hfunc1
hcMGB2.bivar(u1 = c(0.5, 0.1), u2 = c(0.9, 0.1), pars1 = 0.5, pars2 = 0.5, pars3 = 1.5)$hfunc2